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Build up to main theorem
Gross-Tucker Theorem for Labeled Graphs
Coactions and Translation
Fundamental Domain
Labeled graphs C ∗ -algebras with group actions
Teresa Bates, David Pask, Paulette N. Willis∗
University of Iowa
Iowa-Nebraska Functional Analysis Seminar
April 17, 2010
Teresa Bates, David Pask, Paulette N. Willis∗
Labeled graphs C ∗ -algebras with group actions
Build up to main theorem
Gross-Tucker Theorem for Labeled Graphs
Coactions and Translation
Fundamental Domain
Definition
A directed graph E = (E 0 , E 1 , r , s) consists of a vertex set E 0 , an
edge set E 1 , and range and source maps r , s : E 1 → E 0 .
Example
f
e
.
.
v
w
g
Teresa Bates, David Pask, Paulette N. Willis∗
Labeled graphs C ∗ -algebras with group actions
Build up to main theorem
Gross-Tucker Theorem for Labeled Graphs
Coactions and Translation
Fundamental Domain
Definition (Cuntz-Krieger algebra)
One can construct a C ∗ -algebra, C ∗ (E ) with generators {Pv }v ∈E 0
and {Se }e∈E 1 subject to the following relations:
(CK1) The Pv are projections in C ∗ (E ).
(CK2) The Se are partial isometries in C ∗ (E ).
(CK3) Se∗ Se = Ps(e) for all e ∈ E 1 ; and
P
(CK4) Pv = r (e)=v Se Se∗ provided the sum is finite and r −1 (v ) 6= ∅.
The algebra C ∗ (E ) is called a Cuntz-Krieger algebra or a graph
C ∗ -algebra.
The C ∗ -algebra C ∗ (E ) encodes the properties of E , however
non-isomorphic graphs can give rise to isomorphic C ∗ (E ). Graph
C ∗ -algebras are of interest because they are easy to construct and
provide an environment in which to test questions about arbitrary
C ∗ -algebras.
Teresa Bates, David Pask, Paulette N. Willis∗
Labeled graphs C ∗ -algebras with group actions
Build up to main theorem
Gross-Tucker Theorem for Labeled Graphs
Coactions and Translation
Fundamental Domain
Example
Consider the following directed graph
e
.
v
The CK relations say that Se∗ Se = Pv = Se Se∗ . Therefore Pv is the
identity and Se is a unitary operator, hence C ∗ (Se ) is isomorphic
to the continuous functions on the circle C (T).
Teresa Bates, David Pask, Paulette N. Willis∗
Labeled graphs C ∗ -algebras with group actions
Build up to main theorem
Gross-Tucker Theorem for Labeled Graphs
Coactions and Translation
Fundamental Domain
Example
.
e
v
f
The CK relations say that
Se∗ Se = Pv = Sf∗ Sf
and
Pv = Se Se∗ + Sf Sf∗ .
Notice that Pv is the identity for C ∗ (E ). This C ∗ -algebra is simple
(i.e. no non-zero ideals) and is called the Cuntz algebra. It is
denoted O2 .
Teresa Bates, David Pask, Paulette N. Willis∗
Labeled graphs C ∗ -algebras with group actions
Build up to main theorem
Gross-Tucker Theorem for Labeled Graphs
Coactions and Translation
Fundamental Domain
Relationship to dynamical systems
If E = (E 0 , E 1 , r , s) is a finite graph then the infinite path space
E ∞ := {(e1 , e2 , . . .) | s(ei ) = r (ei+1 )}
is a compact subset of (E 1 )N with the product topology that is
invariant under the map σ defined by the equation
σ(e1 , e2 , . . .) := (e2 , e3 , . . .).
The map σ (restricted to E ∞ ) is called a shift of finite type.
Much of the theory of shifts of finite type has direct analogues in
the structure theory of the associated graph C ∗ -algebras.
Teresa Bates, David Pask, Paulette N. Willis∗
Labeled graphs C ∗ -algebras with group actions
Build up to main theorem
Gross-Tucker Theorem for Labeled Graphs
Coactions and Translation
Fundamental Domain
Definition
A labeled graph (E , L) over an alphabet A consists of a directed
graph E together with a labeling map L : E 1 → A.
We assume that L : E 1 → A is surjective, that is L(E 1 ) = A.
Examples
1
A directed graph may be considered a labeled graph where
A = E 1 and the labeling is the identity map.
2
The graph (E , L) over the alphabet {0, 1} is an example of a
labeled graph.
0
f
1 e
.
.
v
w
g
0
Teresa Bates, David Pask, Paulette N. Willis∗
Labeled graphs C ∗ -algebras with group actions
Build up to main theorem
Gross-Tucker Theorem for Labeled Graphs
Coactions and Translation
Fundamental Domain
Example
The graph (E , L) over the alphabet {({0, 1}, Z)} is an example of
a labeled graph, where E 0 = {({v , w }, Z)}.
...
...
(v ,0)
.
(1,0)
(v ,1)
.
(1,1)
(v ,2)
.
(0,0)
(0,1)
(0,2)
(0,0)
(0,1)
(0,2)
(1,2)
(v ,3)
.
...
...
.
.
.
.
(w ,0)
(w ,1)
(w ,2)
(w ,3)
Teresa Bates, David Pask, Paulette N. Willis∗
Labeled graphs C ∗ -algebras with group actions
Build up to main theorem
Gross-Tucker Theorem for Labeled Graphs
Coactions and Translation
Fundamental Domain
Definition
Let e, f ∈ E 1 and s(e) = r (f ), then ef is a path of length 2. E 2 is
the collection of all paths of length 2. Let L(e) = a and L(f ) = b,
then ab ∈ L2 (E ), where L2 (E ) denotes the collection of labels for
paths of length 2. More generally, E n is the collection of paths of
length n and Ln (E ) is the collection of labeled paths of length n.
E ∗ = ∪n∈N E n , and L∗ (E ) = ∪n∈N Ln (E ).
Definition
A vertex v is a sink if it emits no edges, i.e. there does not exist
an edge such that s(e) = v
Definition
For B ⊆ E 0 , let LB := {β ∈ A : B ∩ s(β) 6= ∅} denote the labeled
edges whose source intersects B nontrivially.
Teresa Bates, David Pask, Paulette N. Willis∗
Labeled graphs C ∗ -algebras with group actions
Build up to main theorem
Gross-Tucker Theorem for Labeled Graphs
Coactions and Translation
Fundamental Domain
C ∗ -algebra of labeled graph
Definition
A Cuntz-Krieger (E , L)-family consists of projections
{pr (β) : β ∈ L∗ (E )} and partial isometries {sa : a ∈ A} such that:
(CK1a) pr (β) pr (ω) = 0 if and only if r (β) ∩ r (ω) = ∅
(CK1b) For all β, ω, κ ∈ L∗ (E ), if r (β) ∩ r (ω) = r (κ) then
pr (β) pr (ω) = pr (κ) and if r (β) ∪ r (ω) = r (κ) then
pr (β) + pr (ω) − pr (β) pr (ω) = pr (κ) .
Remark
Using relations (CK1a) and (CK1b) we may now unambiguously
define pr (β)∩r (ω) = pr (β) pr (ω) and
pr (β)∪r (ω) = pr (β) + pr (ω) − pr (β) pr (ω) . If r (β) ∩ r (ω) 6= ∅, then we
write pr (β) pr (ω) = pr (β)∩r (ω) , so (CK1) implies that p∅ = 0.
Teresa Bates, David Pask, Paulette N. Willis∗
Labeled graphs C ∗ -algebras with group actions
Build up to main theorem
Gross-Tucker Theorem for Labeled Graphs
Coactions and Translation
Fundamental Domain
C ∗ -algebra of labeled graph
Definition (continued)
(CK2) If a ∈ A and β ∈ L∗ (E ) then pr (β) sa = sa pr (βa)
(CK3) If a, b ∈ A then sa∗ sa = pr (a) and sa∗ sb = 0 unless a = b
(CK4) For β ∈ L∗ (E ), if Lr (β) is finite, non-empty, and r (β) contains
no sinks we have
X
pr (β) =
sa pr (βa) sa∗ .
a∈Lr (β)
Teresa Bates, David Pask, Paulette N. Willis∗
Labeled graphs C ∗ -algebras with group actions
Build up to main theorem
Gross-Tucker Theorem for Labeled Graphs
Coactions and Translation
Fundamental Domain
Relationship to dynamical systems
Recall that the theory of shifts of finite type has direct analogues
in the theory of graph C ∗ -algebras.
Labeled graph C ∗ -algebras are similarly connected to sofic shifts.
Teresa Bates, David Pask, Paulette N. Willis∗
Labeled graphs C ∗ -algebras with group actions
Build up to main theorem
Gross-Tucker Theorem for Labeled Graphs
Coactions and Translation
Fundamental Domain
Definition
Let (E , L) and (F , M) be labeled graphs over alphabets AE and
AF respectively. A labeled graph isomorphism is a triple
φ := (φ0 , φ1 , φA ) where φ0 : E 0 → F 0 , φ1 : E 1 → F 1 , and
φAE : AE → AF such that
1
2
3
φ0 , φ1 , φAE are bijective
For all e ∈ E 1 we have φ0 (r (e)) = r (φ1 (e)) and
φ0 (s(e)) = s(φ1 (e));
φAE : AE → AF is a map such that M ◦ φ1 = φAE ◦ L.
If F = E and AE = AF , then φ is called a labeled graph
automorphism.
Let (E , L) be a labeled graph over the alphabet A, then the set
Aut(E , L) := {φ : (E , L) → (E , L) : φ is a labeled graph automorphism}
forms a group under composition.
Teresa Bates, David Pask, Paulette N. Willis∗
Labeled graphs C ∗ -algebras with group actions
Build up to main theorem
Gross-Tucker Theorem for Labeled Graphs
Coactions and Translation
Fundamental Domain
Definition (group action)
Let (E , L) be a labeled graph over the alphabet A and G be a
group. A labeled graph action of G on (E , L) is a triple
((E , L), G , α) where α : G → Aut(E , L) is a group homomorphism.
So for all e ∈ E 1 and g ∈ G we have L(αg1 (e))=αgA (L(e)).
E1
.
α1g
L
.
A
E1
.
L
αA
g
.
A
We say that the label graph action ((E , L), G , α) is free if
αg0 (v ) = v for all v ∈ E 0 implies g = 1G and if αgA (a) = a for all
a ∈ A then g = 1G .
Teresa Bates, David Pask, Paulette N. Willis∗
Labeled graphs C ∗ -algebras with group actions
Build up to main theorem
Gross-Tucker Theorem for Labeled Graphs
Coactions and Translation
Fundamental Domain
Definition (quotient labeled graph)
Let ((E , L), G , α) be a labeled graph action. For i = 0, 1, x ∈ E i ,
and a ∈ A let
Ga = {αgA (a) : g ∈ G }
Gx := {αgi (x) : g ∈ G }
(E /G )i = {Gx : x ∈ E i }
1
L/G : (E /G ) → A/G
A/G = {Ga : a ∈ A}
be given by
r (Ge) = Gr (e)
(L/G )(Ge) = G L(e)
s(Ge) = Gs(e)
for Ge ∈ (E /G )1 . Then (E /G , L/G ) is a labeled graph over A/G
which we call the quotient labeled graph.
Teresa Bates, David Pask, Paulette N. Willis∗
Labeled graphs C ∗ -algebras with group actions
Build up to main theorem
Gross-Tucker Theorem for Labeled Graphs
Coactions and Translation
Fundamental Domain
Example
Consider the following labeled graph
...
v0
.
v1
10
00
12
.
01
00
...
v2
11
.
v3
.
...
...
02
01
02
.
.
.
.
w0
w1
w2
w3
where E 0 = {{vi , wi } : i ∈ Z}, A = {{0i , 1i } : i ∈ Z}, G = Z, α is
left translation. Then the quotient labeled graph is
0
1
.
.
v
w
0
Teresa Bates, David Pask, Paulette N. Willis∗
Labeled graphs C ∗ -algebras with group actions
Build up to main theorem
Gross-Tucker Theorem for Labeled Graphs
Coactions and Translation
Fundamental Domain
Main Theorem (Bates, Pask, W*)
Let (E , L) be a labeled graph and let G be a group acting freely
on (E , L) via α. Then (under a technical hypothesis) G acts on
C ∗ (E , L) via α
e and
C ∗ (E , L) ×αe,r G ∼
= C ∗ (E /G , L/G ) ⊗ K(`2 (G )),
i.e. C ∗ (E , L) ×αe,r G is strongly Morita equivalent to
C ∗ (E /G , L/G ).
Teresa Bates, David Pask, Paulette N. Willis∗
Labeled graphs C ∗ -algebras with group actions
Build up to main theorem
Gross-Tucker Theorem for Labeled Graphs
Coactions and Translation
Fundamental Domain
Steps
To prove
C ∗ (E , L) ×αe,r G ∼
= C ∗ (E /G , L/G ) ⊗ K(`2 (G ))
we need two theorems.
The Gross-Tucker theorem for labeled graphs: a free action on
a labeled graph is naturally equivariantly isomorphic to a skew
product action obtained from the quotient labeled graph.
A generalization of a theorem of Kaliszewski, Quigg, and
Raeburn: the C ∗ -algebra of a skew product labeled graph is
naturally isomorphic to a co-crossed product of a coaction of
the group on the C ∗ -algebra of the labeled graph.
Teresa Bates, David Pask, Paulette N. Willis∗
Labeled graphs C ∗ -algebras with group actions
Build up to main theorem
Gross-Tucker Theorem for Labeled Graphs
Coactions and Translation
Fundamental Domain
While we are interested in skew products of quotient graphs, for
now we will define the skew product of a general labeled graph.
Definition
Let (E , L) be a labeled graph and let c, d : E 1 → G be functions.
The skew product labeled graph (E ×c G , Ld ) over alphabet
A × G is the skew product graph E ×c G defined by:
(E ×c G )0 := E 0 × G
r (e, g ) := (r (e), gc(e))
(E ×c G )1 := E 1 × G
s(e, g ) := (s(e), g )
together with the labeling Ld : E 1 × G → A × G given by
Ld (e, g ) := (L(e), gd(e)).
Teresa Bates, David Pask, Paulette N. Willis∗
Labeled graphs C ∗ -algebras with group actions
Build up to main theorem
Gross-Tucker Theorem for Labeled Graphs
Coactions and Translation
Fundamental Domain
Example
Consider the labeled graph (E , L)
0
.
1
.
v
w
0
Let G = Z and c, d : E 1 → Z be c(ε) = 1 and d(ε) = 0, for
ε ∈ E 1 . Then the skew-product labeled graph (E ×c Z, Ld ) is:
...
...
(v ,0)
.
(1,0)
(v ,1)
.
(1,1)
(v ,2)
.
(0,0)
(0,1)
(0,2)
(0,0)
(0,1)
(0,2)
(1,2)
(v ,3)
.
...
...
.
.
.
.
(w ,0)
(w ,1)
(w ,2)
(w ,3)
Teresa Bates, David Pask, Paulette N. Willis∗
Labeled graphs C ∗ -algebras with group actions
Build up to main theorem
Gross-Tucker Theorem for Labeled Graphs
Coactions and Translation
Fundamental Domain
Example
For (x, h) ∈ (E ×c G )i , (a, h) ∈ A × G , g ∈ G , and i = 0, 1 let
τgi (x, h) = (x, gh)
and
τgA (a, h) = (a, gh).
The map τ = (τ 0 , τ 1 , τ A ) : G → Aut(E ×c G , Ld ) defined by
g → τg is called the (left) labeled graph translation map.
The left labeled graph translation action is a free action since G
acts freely on itself by left translation.
Teresa Bates, David Pask, Paulette N. Willis∗
Labeled graphs C ∗ -algebras with group actions
Build up to main theorem
Gross-Tucker Theorem for Labeled Graphs
Coactions and Translation
Fundamental Domain
Gross-Tucker Theorem for Labeled Graphs
Theorem (Bates, Pask, W*)
Let ((E , L), G , α) be a free labeled graph action. There are
functions c, d : (E /G )1 → G such that ((E , L), G , α) is
equivariantly isomorphic to ((E /G ×c G , (L/G )d ), G , τ ), i.e. the
isomorphism intertwines the actions.
So by this theorem we have
C ∗ (E , L) ×αe,r G ∼
= C ∗ (E /G ×c G , (L/G )d ) ×τe,r G .
Teresa Bates, David Pask, Paulette N. Willis∗
Labeled graphs C ∗ -algebras with group actions
Build up to main theorem
Gross-Tucker Theorem for Labeled Graphs
Coactions and Translation
Fundamental Domain
Example
v
0
... .
(E ,L):=
v1
10
v2
11
.
.
00
v3
12
. ...
02
01
... .
w0
.
.
w1
w2
. ...
w3
0
(E /Z,L/Z):=
.
1
.
v
w
0
(v ,0)(1,0)(v ,1)(1,1)(v ,2)(1,2)(v ,3)
... .
(E /Z×c Z,Ld ):=
(0,0)
... .
(w ,0)
Teresa Bates, David Pask, Paulette N. Willis∗
.
(0,1)
.
. ...
(0,2)
.
.
(w ,1)
(w ,2)
. ...
(w ,3)
Labeled graphs C ∗ -algebras with group actions
Build up to main theorem
Gross-Tucker Theorem for Labeled Graphs
Coactions and Translation
Fundamental Domain
Definition
Let (E , L) be a labeled graph and G be a group. A function
c : E 1 → G is label consistent if whenever e, f ∈ E 1 satisfy
L(e) = L(f ), then c(e) = c(f ).
Example
Consider the labeled graph (E , L)
1 e
.
0
f
v
.
w
g
0
Let G = Z and c, d : E 1 → Z be c(ε) = 1 and d(ε) = 0, ∀ε ∈ E 1 .
If c : E 1 → G is label consistent function, it induces a well-defined
function C : A → G such that C (a) = c(e) where L(e) = a.
Teresa Bates, David Pask, Paulette N. Willis∗
Labeled graphs C ∗ -algebras with group actions
Build up to main theorem
Gross-Tucker Theorem for Labeled Graphs
Coactions and Translation
Fundamental Domain
Lemma
Let (E , L) be a labeled graph, G be a discrete group, and
c : E 1 → G be a label consistent function.Then there is a coaction
δ : C ∗ (E , L) → C ∗ (E , L) ⊗ C ∗ (G ) such that
δ(sa ) = sa ⊗ uC (a)
and
δ(pr (β) ) = pr (β) ⊗ u1G
where {sa , pr (β) } is a Cuntz-Krieger (E , L)-family and
{ug : g ∈ G } are the generators of C ∗ (G ).
Teresa Bates, David Pask, Paulette N. Willis∗
Labeled graphs C ∗ -algebras with group actions
Build up to main theorem
Gross-Tucker Theorem for Labeled Graphs
Coactions and Translation
Fundamental Domain
Remark
Recall ((E ×c G , Ld ), G , τ ), the left labeled graph translation
action, which is a free action. This induces an action
τe : G → Aut C ∗ (E ×c G , Ld ) such that
τeh (sa,g ) = sa,hg
and
τeh (pr (β),g ) = pr (β),hg .
Remark
The C ∗ -algebra C ∗ (E , L) ×δ G carries an action δb of G defined via
the formula δbh (bg , x) = (bg , xh−1 ).
Teresa Bates, David Pask, Paulette N. Willis∗
Labeled graphs C ∗ -algebras with group actions
Build up to main theorem
Gross-Tucker Theorem for Labeled Graphs
Coactions and Translation
Fundamental Domain
Theorem (Bates, Pask, W*)
Let (E , L) be a labeled graph, let G be a discrete group, let
c, d : E 1 → G be label consistent functions, and let δ be the
coaction. Then
C ∗ (E ×c G , Ld ) ∼
= C ∗ (E , L) ×δ G
b
equivariantly for the action τe and the dual action δ.
Remark
C ∗ (E ×c G ) ∼
= C ∗ (E ) ×δ G was proven by Kaliszewski, Quigg, and
Raeburn, so our result that C ∗ (E ×c G , Ld ) ∼
= C ∗ (E , L) ×δ G is a
labeled graph version of their result.
Teresa Bates, David Pask, Paulette N. Willis∗
Labeled graphs C ∗ -algebras with group actions
Build up to main theorem
Gross-Tucker Theorem for Labeled Graphs
Coactions and Translation
Fundamental Domain
Since the isomorphism is equivariant for τe and δb it follows that
C ∗ (E ×c G , Ld ) ×τe,r G ∼
= C ∗ (E , L) ×δ G ×b G .
δ,r
By Katayama’s duality theorem we know that
C ∗ (E , L) ×δ G ×b G ∼
= C ∗ (E , L) ⊗ K(`2 (G )).
δ,r
Therefore we have
Theorem
Let (E , L) be a labeled graph, let G be a discrete group, let
c, d : E 1 → G be label consistent functions, and let τ be the left
labeled translation map. Then
C ∗ (E ×c G , Ld ) ×τe,r G ∼
= C ∗ (E , L) ⊗ K(`2 (G )).
If the labeled graph is a quotient labeled graph (E /G , L/G ) then
C ∗ (E /G ×c G , (L/G )d ) ×τe,r G ∼
= C ∗ (E /G , L/G ) ⊗ K(`2 (G )).
Teresa Bates, David Pask, Paulette N. Willis∗
Labeled graphs C ∗ -algebras with group actions
Build up to main theorem
Gross-Tucker Theorem for Labeled Graphs
Coactions and Translation
Fundamental Domain
The Gross-Tucker theorem for labeled graphs proves
((E , L), G , α) ∼
= ((E /G ×c G , (L/G )d ), G , τ )
which gives us C ∗ (E , L) ×αe,r G ∼
= C ∗ (E /G ×c G , (L/G )d ) ×τe,r G .
We have also shown that
C ∗ (E /G ×c G , (L/G )d ) ×τe,r G ∼
= C ∗ (E /G , L/G ) ⊗ K(`2 (G )),
therefore we have
C ∗ (E , L) ×αe,r G ∼
= C ∗ (E /G , L/G ) ⊗ K(`2 (G )).
Remark
Notice that to define the coaction δ we relied on the assumption
that c is label consistent. We need d label consistent to identify
L∗d (E ×c G ) with L∗ (E ) × G . Since the Gross-Tucker theorem only
gives the existence of functions c and d, we must determine
conditions under which c and d are label consistent.
Teresa Bates, David Pask, Paulette N. Willis∗
Labeled graphs C ∗ -algebras with group actions
Build up to main theorem
Gross-Tucker Theorem for Labeled Graphs
Coactions and Translation
Fundamental Domain
Definition
Let ((E , L), G , α) be a free labeled graph action. A fundamental
domain for ((E , L), G , α) is a set T ⊆ E 0 such that
1
for every v ∈ E 0 there exists g ∈ G and a unique w ∈ T such
that v = αg0 w ,
2
if r (e), r (f ) ∈ T , and G L(e) = G L(f ), then L(e) = L(f ),
3
if s(e), s(f ) ∈ T , and G L(e) = G L(f ), then L(e) = L(f )
for every e, f ∈ E 1 .
Proposition
Let ((E , L), G , α) be a free labeled graph action. Then by
Gross-Tucker for labeled graphs there exists functions
c, d : (E /G )1 → G such that
((E , L), G , α) ∼
= ((E /G ×c G , (L/G )d ), G , τ ). If ((E , L), G , α)
admits a fundamental domain, then c and d are label consistent.
Teresa Bates, David Pask, Paulette N. Willis∗
Labeled graphs C ∗ -algebras with group actions
Build up to main theorem
Gross-Tucker Theorem for Labeled Graphs
Coactions and Translation
Fundamental Domain
Theorem (Bates, Pask, W*)
Let (E , L) satisfy the nondegenerate hypothesis and ((E , L), G , α)
be a free labeled graph action which admits a fundamental
domain. Then
C ∗ (E , L) ×αe,r G ∼
= C ∗ (E /G , L/G ) ⊗ K(`2 (G )).
Proof.
There are label consistent functions c, d : (E /G )1 → G such that
((E , L), G , α) ∼
= ((E /G ×c G , (L/G )d ), G , τ ),
so we have C ∗ (E , L) ×αe,r G ∼
= C ∗ (E /G ×c G , (L/G )d ) ×τe,r G .
We have shown that
C ∗ (E /G ×c G , (L/G )d ) ×τe,r G ∼
= C ∗ (E /G , L/G ) ⊗ K(`2 (G )).
Teresa Bates, David Pask, Paulette N. Willis∗
Labeled graphs C ∗ -algebras with group actions